
where \(\alpha \)  and \(\beta ^i\)  are the lapse and shift respectively.
This decomposition introduces spatial hypersurfaces characterized by the
normal vector \(n_\mu = \left(-\alpha ,0,0,0\right) \) .
Within this work, we will adopt a corotating flat frame, in which
\(\alpha =1\)  and \(\gamma _{ij} = \delta _{ij}\) , but non-zero shift \(\beta ^k\) . We will specify the precise
choice of \(\beta ^k\)  in Sec. REF , see also [1]}, [2]}.
The dynamics of electromagnetic fields in relativity is governed by the
generalized form of Maxwell's equations
[3]},
\(\nabla _\mu \left(F^{\mu \nu } + \phi g^{\mu \nu }\right) &= - 4 \pi \mathcal {J}^\nu +\kappa _\phi n^\nu \phi ,  \\\nabla _\mu \left(^\ast F^{\mu \nu } +\psi g^{\mu \nu }\right) &=+\kappa _\psi n^\nu \psi \,, \\\nabla _\mu \mathcal {J}^\mu &=0\,.\) 
